Lilliefors Test of Normality

Test for class of normal distributions.

We can write the class or set of normal distributions as:

{N(μ,σ2)∣μ∈R,σ2>0}\left\{N\left(\mu, \sigma^2\right) \mid \mu \in \mathbb{R}, \sigma^2>0\right\}

Our Null Hypothesis thus is

H0:F∈{N(μ,σ2)∣μ∈R,σ2>0}H_0:F\in \left\{N\left(\mu, \sigma^2\right) \mid \mu \in \mathbb{R}, \sigma^2>0\right\}

or in words, "is the CDF of our data sequence in the class of normal distributions?"

We define our Test Statistic LnL_n as the supremum of the absolute distances between the ECDF and the CDF of a normal distribution with estimated Arithmetic Mean and Variance.

The distribution of LnL_n is independent of the Variance and Expectation of the Normal Distribution and may thus be simulated.

To make up for the increasing nn we use the square root of it to scale the distribution.

We accept H0H_0:

  • if the test statistic is ≤L1−αn\leq \frac{L_{1-\alpha}}{\sqrt{n}}
  • if α<p(l)\alpha < p(l) with p(l)=1−L(nl)p(l)=1-L(\sqrt{n}l) the p-Value of the test.

Usage in R

#todo